Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision | |||
|
pullback_._category_theory [2015/03/16 20:39] nikolaj |
pullback_._category_theory [2015/03/16 21:06] nikolaj |
||
|---|---|---|---|
| Line 27: | Line 27: | ||
| == Special cases == | == Special cases == | ||
| * If $\pi_a$ is an iso, then $A\times_Z B\cong A$. As $A$ is already the pullback, it alone fully determines the "full solution". | * If $\pi_a$ is an iso, then $A\times_Z B\cong A$. As $A$ is already the pullback, it alone fully determines the "full solution". | ||
| - | * If moreover $\pi_b$ is an iso too, the projections vanish from the diagram and the universal property says that arrows $\gamma,\delta$ (see above) can be wholly glued together, i.e., up to iso, $\alpha\circ\gamma=\beta\circ\delta\implies\gamma=\delta$. | + | * If moreover $\pi_b$ is an iso too, the projections, we can consider the equivalent pullback with $\pi_b=\pi_a=1_A$. The universal property now says that arrows $\gamma,\delta$ can be wholly glued together: Up to iso, $\alpha\circ\gamma=\beta\circ\delta\implies\gamma=\delta$. |
| - | * In ${\bf{Set}}$, if $\alpha=\beta$, the pullback definition says its elements $\langle x,y\rangle$ fulfill $\alpha(x)=\alpha(y)$, i.e. here the pullback object is the full collection of pairs of term with give the same $\alpha$ value. If moreover $\pi_a$ is iso, any $x$ determines an $\langle x,y\rangle$ and hence a $y$ and the universal property says $\alpha(x)=\alpha(y)\implies x=y$. This is just the definition of an injection. | + | * In ${\bf{Set}}$, if $\alpha=\beta$, the pullback definition says that its elements $\langle x,y\rangle$ fulfill $\alpha(x)=\alpha(y)$, i.e. here the pullback object is the full collection of pairs of term with give the same $\alpha$ value. If moreover $\pi_a$ is iso, any $x$ determines an $\langle x,y\rangle$ and hence an $y$ and the universal property translates to $\alpha(x)=\alpha(y)\implies x=y$. This is just the definition of an injection. |
| - | * Back to a general category, consider the case where $\pi_a$ is iso AND $\alpha=\beta$. The condition is $\alpha\circ\gamma=\alpha\circ\delta\implies\gamma=\delta$ and we call such an $\alpha$ a [[monomorphism]] | + | * Back to a general category. If the pullback of $\alpha$ along itself ($\alpha=\beta$) is such that a projection $\pi_a$ is iso, we call $\alpha$ a [[monomorphism]]. The associated condition reads $\alpha\circ\gamma=\alpha\circ\delta\implies\gamma=\delta$. |
| {{ monomorphism-diagram.png?X300}} | {{ monomorphism-diagram.png?X300}} | ||
